A 300 room hotel is twothirds filled when the nightly room r

A 300- room hotel is two-thirds filled when the nightly room rate is $90. Experience has shown that each $5 increase in cost results in 10 fewer occupied rooms. Find the nightly rate that will maximize income.

Solution

Given, two-thirds filled when the nightly room rate is $90.

As, there are total 300 rooms, two thirds of it would be 200.

So, 200 rooms are occupied when the nightly rate is $90.

Let \'x\' be the number of $5 increased nightly rooms.

Then its given, it results in 10 fewer occupied rooms.

So, the nightly rate would be \"90 + 5x\" and the number of occupied rooms would be \"200 - 10x\"

Then the, income would be, (90 + 5x)(200 - 10x)

Let the income be denoted by \'y\'

Then we have y = (90 + 5x)(200 - 10x)

Which can be written as, y = 18000 - 900 x + 1000 x - 50 x²

That is, y = -50 x^2 + 100x + 18000, which is a parabola opening downwards.

For which, the maximum would be attained at the vertex.

The x co ordinate of the vertex of the parabola is given by, x = - b / 2a

That is, x = - 100 / 2(-50) = (-100)/(-100) = 1

So for x = 1, y attains maximum.

For x = 1, the nightly rate \"90 + 5x\" becomes 90 + 5(1) = 95.

That is, for the nightly rate $95, the income would be maximum.